I see where you're having a hard time explaining these two sentences to native Portuguese speakers. The things is that in Portuguese "I don't want anything" and "I want nothing" has the same translation. If you say "eu quero nada" to Portuguese speakers they will understand, but it doesn't sound natural at all, that is why "Eu não quero nada" stands for both sentences. In Portuguese, double negatives are grammatically correct and we use them a lot, so a sentence like "I don't know anything" and "I know nothing" has the same translation into Portuguese "Eu não sei de nada". I don't know if I got the message across so far haha. You can try explaining to them that if they say "I want nothing" means that they actually want nothing, they would give away all that they have in order to have nothing at all. Whereas, "I don't want anything" implies that they don't want more than they already have at the moment, which is what you explained in your question. The hard one to explain is " I want nothing". So, "I don't want to do anything " or "I want to do nothing " (Eu não quero fazer nada) you can explain that the first one, the person really wants to do nothing, he or she doesn't want to do any other activity besides nothing and the second one, the person doesn't want to do what someone expects he or she to do or what he or she has to do, but they will eventually do other thing. It is tough to explain this because literally both sound the same to us. In Portuguese, it doesn't make sense to want nothing, how do you want nothing if nothing is not something? haha.
I hope I was able to help you!